Arbitrage question

Question

Consider a hypothetical Payment in Kind (PIK) bond of XYZ Corporation. The bond has 2 years to maturity, a face value of $1000, and has an annual coupon rate of 10%.

Coupons are paid annually. XYZ has the right to pay the first coupon either in cash or in additional PIK bonds – i.e., the bond holder may get either $100 in cash or 10 additional PIK bonds for every 100 bonds she has. The second coupon must however be paid in cash along with the face value at the end of two years. The PIK bonds are risk free and trade at par while the yield curve is flat at 9%

risk free 1 year and 2 year zero coupon bonds trade at a yield to maturity of 9% (Effective Annual Yield).

Suppose you can buy and short sell (borrow and sell) the PIK and zero coupon bonds without transactions costs. You forecast that the yield to maturity on one year zero coupon bonds one year from now will either stay at 9% or change to 8.5% or 9.5%.

a) Assume XYZ always pays the first coupon in cash, what should be the price of the bond?

b) Given your forecast on future interest rate (as stated in the problem), show that there is an arbitrage opportunity. (

Answer 1

First, set up the initial position with 0 initial cost: buy 1 PIK bond at face value (1000\$) and sell 1000$ worth of the 2-year zero-coupon to finance the purchase.

After one year, you either get paid 0.1 PIK bonds or 100$. In the first case, the position is kept unchanged until maturity at year 2. Your payoff will be the repayment of the PIK bonds (+ coupon) - the value of the zero bonds:

$(1) V_2 = 1.1 V_{PIK} - 1000 (1+0.09)^2 = 1.1 (1000 + 100) - 1000 (1.09)^2 = 1210 - 1188.1 = 21.90$

For the second scenario, you get 100$ from the coupon, which you reinvest in a 1-year ZC. If the worst-case scenario (lowest reinvestment rate, which is 8.5% in this case) yields a profit, we'll have an arbitrage. So, at maturity, we get:

$(2) V_2 = 1 V_{PIK} - 1000 (1+0.09)^2 + 100 (1+0.085) = 1 (1000 + 100) - 1000 (1.09)^2 + 100 (1.085) = 1100 - 1188.1 + 108.5 = 20.40$

We've just proved that a zero-cost portfolio yields a positive profit under all scenarios, which is the definition of an arbitrage.

Another possible arbitrage can be built by financing the buying of the PIK bond by selling $100\$ (1+0.09)^{-1} = 91.74$ of the 1-year ZC, and making up the rest (908.26) by selling the 2-year ZC. This is also a zero-cost portfolio; but, in this case, if the year 1 coupon is in cash, we get repay the borrowing earlier.

At year 1, same scenarios as before. But, in this case, if we receive 0.1 PIK, we need to roll over the short-term borrowing; so we'll use the worst-case rate for borrowing, which is 9.5%. At year 2, we have:

$(1) V_2 = 1.1 V_{PIK} - (908.26)(1.09)^2 - 100(1.095) = 1210 - 1079.10 - 109.50 = 1210 - 1188.60 = 21.40$

If the coupon is in cash, we repay the 100$ worth of the 1-year ZC, and at year 2, we have:

$(2) V_2 = 1 V_{PIK} - (908.26)(1.09)^2 = 1100 - 1079.10 = 20.90$

Again, the zero-cost initial portfolio yields a positive profit with probability 1, so constitutes an arbitrage.

Answer 2

a) $$P_t=100/1.09+(100+1000)/1.09^2=1017,591112$$